Channel-parameter estimation for satellite-to-submarine continuous-variable quantum key distribution

This paper deals with a channel-parameter estimation for continuous-variable quantum key distribution (CV-QKD) over a satellite-to-submarine link. In particular, we focus on the channel transmittances and the excess noise which are affected by atmospheric turbulence, surface roughness, zenith angle of the satellite, wind speed, submarine depth, etc. The estimation method is based on proposed algorithms and is applied to low-Earth orbits using the Monte Carlo approach. For light at 550 nm with a repetition frequency of 1 MHz, the effects of the estimated parameters on the performance of the CV-QKD system are assessed by a simulation by comparing the secret key bit rate in the daytime and at night. Our results show the feasibility of satellite-to-submarine CV-QKD, providing an unconditionally secure approach to achieve global networks for underwater communications.

Figure 1

(a) The scheme of CV-QKD over a satellite-to-submarine link. Modulated light is transmitted through the atmosphere, sea surface, and seawater, in that order. With the LO, which is generated by a local laser, Bob (submarine vehicle) measures the states using homodyne detection. (b) The top and bottom figures represent the atmospheric transmittance and clear ocean water attenuation as functions of wavelength, respectively, which are both incurred by absorption and scattering. AM: amplitude modulator; PM: phase modulator; PD: photodiode; Tele.: telescope; Att.: attenuator; LO: local oscillator; $\zeta$: zenith angle of the satellite.

Figure 2

(a) The refraction structure $C_n^2$ at altitudes of 0, 1, 10, and 20 km with various wind speeds and constant parameter $A$. (b) The scintillation index as a function of the zenith angle of a satellite at altitudes from 200 to 40 000 km.

Figure 3

(a) The derivation process of atmospheric transmittance and (b) the aperture of radius $r_0$ and the elliptical beam profile with the half axis $W_1$ and $W_2$, where $W_1$ rotates on the angle $\mathrm{\Phi }$ relating to the $x$ axis. Beam wandering is characterized by parameter $r$, which represents the beam-centroid position with respect to the center of the aperture.

Figure 4

The probability density distribution of atmospheric transmittance with various zenith angles. The low-Earth orbits of the satellite are set to (a) 200, (b) 400, (c) 600, and (d) 800 km. The dashed lines represent the maximal zenith angle at the corresponding altitudes.

Figure 5

The geometric model of the satellite, sea surface, and receiver plane. The top dotted circle contains the possible positions of the satellite at certain altitudes and zenith angle $\zeta$. In the magnification of the local sea surface, $\varepsilon$ corresponds to the wave slope while ${\theta }_i$ and ${\theta }_r$ represent the light incident angle and refraction angle, respectively.

Figure 6

The light intensity distribution derived from a rough sea surface at different depths of water. The zenith angles of the satellite are set to (a) $0^{\circ }$, (b) ${20}^{\circ }$, (c) ${40}^{\circ }$, and (d) ${60}^{\circ }$, respectively. The total number of photons for the Monte Carlo simulation is ${10}^6$ for each situation and the wind speed is 3 m/s.

Figure 7

The percentage of received photons as a function of different wind speeds (3, 6, 9, and 12 m/s) and depths (20, 40, 60, and 80 m). The result is based on the assumption that the zenith angle $\zeta =0^{\circ }$. The number of photons used for the Monte Carlo simulation is ${10}^6$ for each case. The radius of the receiver is 1 m.

Figure 8

The transmittance probability distribution of a rough sea surface based on different wind speeds: (a) 3, (b) 6, (c) 9, and (d) 12 m/s. The number of photons used for the Monte Carlo simulation is ${10}^6$ for each case.

Figure 9

(a) An ideal model of light propagation without other turbulence except intensity loss. (b) The ENL in shot-noise units (SNU). (c) The background noise power received by a virtual receiver at the surface as a function of brightness and field of view. (d) The power of background noise underwater as a function of depth and brightness. The excess noise of the initial states is 0.01 SNU. The parameters are set to $W_0=6\mathrm{cm}$, $r_0=1\mathrm{m}$, $R=1.25\%$, $c=0.166{\mathrm{m}}^{-1}$, and $L_{\mathrm{fac}}=1$.

Figure 10

The worst total excess noise (maximal zenith angle and ${\mathrm{\Omega }}_{\mathrm{fov}}=\pi$) in (a) clear daytime and (b) on a moonless, clear night, respectively. Six cases at different altitudes (from $H=200$ to 1200 km in steps of 200 km) and seven depths (from $D=20$ to 80 m in steps of 10 m) are investigated. The dashed lines at the $xOz$ axis plane represent the ENL at their corresponding altitudes.

Figure 11

The equivalent theoretical model of the satellite-to-submarine CV-QKD with Gaussian-modulated states. $T_{\mathrm{air},i}$: atmospheric transmittance; $T_{\mathrm{surface},i}$: transmittance of the fluctuating sea surface; $T_{\mathrm{sea}}$: underwater transmittance at a certain depth; $\chi$: total excess noise; ${\eta }_{\mathrm{B}}$ and $v_{\mathrm{el}}$: the efficiency and electrical noise of the homodyne detector, respectively.

Figure 12

The secret key bit rate as a function of altitude and depth based on a flat sea surface (i.e., wind speed $v=0$ m/s and thus ${\eta }_{\mathrm{deflection}}=1$). The altitudes of the satellite are set to (a) 200, (b) 400, (c) 600, and (d) 800 km.

Figure 13

The bit rate of the secret key (a) in the daytime and (b) at night based on different wind speeds at an altitude of 200 km.